You have found the following ages (in years) of all $5$ porcupines at your local zoo: $ 16,\enspace 10,\enspace 5,\enspace 7,\enspace 13$ What is the average age of the porcupines at your zoo? What is the standard deviation? Round your answers to the nearest tenth. Average age: $ $
Answer: Because we have data for all $5$ porcupines at the zoo, we are able to calculate the population mean $({\mu})$ and population standard deviation $({\sigma})$. To find the population mean, add up the values of all $5$ ages and divide by $5$. $ {\mu} = \dfrac{\sum\limits_{i=1}^{{N}} x_i}{{N}} = \dfrac{\sum\limits_{i=1}^{{5}} x_i}{{5}} $ $ {\mu} = \dfrac{16 + 10 + 5 + 7 + 13}{{5}} = {10.2\text{ years old}} $ Find the squared deviations from the mean for each porcupine. Age $x_i$ Distance from the mean $(x_i - {\mu})$ $(x_i - {\mu})^2$ $16$ years $5.8$ years $33.64$ years $^2$ $10$ years $-0.2$ years $0.04$ years $^2$ $5$ years $-5.2$ years $27.04$ years $^2$ $7$ years $-3.2$ years $10.24$ years $^2$ $13$ years $2.8$ years $7.84$ years $^2$ Because we used the population mean $({\mu})$ to compute the squared deviations from the mean, we can find the variance $({\sigma^2})$, without introducing any bias, by simply averaging the squared deviations from the mean: $ {\sigma^2} = \dfrac{\sum\limits_{i=1}^{{N}} (x_i - {\mu})^2}{{N}} $ $ {\sigma^2} = \dfrac{{33.64} + {0.04} + {27.04} + {10.24} + {7.84}} {{5}} $ $ {\sigma^2} = \dfrac{{78.8}}{{5}} = {15.76\text{ years}^2} $ As you might guess from the notation, the population standard deviation $({\sigma})$ is found by taking the square root of the population variance $({\sigma^2})$. ${\sigma} = \sqrt{{\sigma^2}}$ $ {\sigma} = \sqrt{{15.76\text{ years}^2}} = {4\text{ years}} $ The average porcupine at the zoo is $10.2$ years old. There is a standard deviation of $4$ years.